A Simple Model of Intermediation

Banks in our framework design loan contracts and monitor borrower behaviour. Such actions aim to curb borrowers’ moral hazard resulting in tendencies towards strategic default or outright repudiation on borrowed funds. Loan contract design and borrower monitoring have provided a unifying theme in models of financial intermediation (Diamond 1984; Rajan 1992; Besanko and Kanatas 1993; Holmström and Tirole 1997; Repullo and Suarez 1998).

We present a one-period model of financial intermediation with a single representative bank that performs tasks as an active lender and passive holder of deposits. The bank operates in a competitive market for deposits. Deposits are used to finance loans to individual borrowers. While the bank pays a competitive rate to depositors, it decides upon loan size, loan rate and the effort devoted to the ex-post monitoring of borrowers.

The ex-post monitoring of borrowers is costly, but reduces the probability of loan default. The bank’s monitoring effort reduces the risk of loan default, and leads to a decline in the spread between deposit and loan interest rates. The model posits that if a tax is levied on the profit the bank earns from offering financial intermediation services to borrowers and depositors, then such a tax affects directly core financial intermediation activities including the volume of loans and deposits, and the interest rates for depositors and borrowers.

The bank engages with both borrowers and depositors via a set of loan and deposit contracts. In the remainder of this section, we present a model which addresses how key contractual variables, such as the size of loans and deposits,

59

loan and deposit rates, and monitoring effort are affected by a sudden increase in tax. For the purposes of exposition, we assume that depositors and borrowers are two distinct sets of agents. This allows us to analyse the features of loan and deposit contracts separately, before combining these to examine the overall impact of taxes on financial intermediation.

Loan Contracts – Borrowers

Each borrower has a project which produces a cash flow with a technology given by a concave production function,𝑓(𝐿), where𝐿 denotes the loan amount. We

impose the following assumption on the technology: 𝑓/_{(𝐿) > 0 and𝑓}//_{(𝐿) < 0. An} example of such a technology is 𝑓(𝐿) = 𝐴√𝐿, where 𝐴 is a parameter. Borrowers

do not have any internal means of finance, and so rely on bank financing. The bank charges interest rate𝑅 against a loan amount 𝐿. The bank also chooses the

probability, 𝑝, of monitoring each borrower in order to deter strategic default.

Given the one-period nature of the model capturing the relationship between the borrowers and the bank, there is no scope for reputation building by the borrower (which would emerge from repeated interactions). Hence, borrowers are more likely to default strategically after securing financing. Financial intermediation and lending in particular is special in this context, since banks can use information and expertise to monitor borrowers in order to deter strategic default.

A borrower may or may not behave honestly depending on the payoff (gains and costs) associated with such behaviours. If the bank charges a loan rate

𝑅, on a loan amount 𝐿, disbursed to a borrower, the pay-off to an honest borrower

(who repays the total loan obligation) is 𝑓(𝐿) − 𝑅𝐿. Whether a borrower repays a

60

dishonestly, then a cost is incurred which takes a fraction 𝛼 of output 𝑓(𝐿). If the

borrower gets caught by the bank, 𝑅𝐿 is paid back and legal and other pecuniary

expenses amounting to 𝑐are incurred. The borrower’s expected pay-off from

dishonest behaviour is 𝑝[𝛼𝑓(𝐿) − 𝑅𝐿 − 𝑐] + (1 − 𝑝)𝑓(𝐿). Hence the borrower’s

incentive compatibility condition is 𝑓(𝐿) − 𝑅𝐿 ≥ 𝑝[𝛼𝑓(𝐿) − 𝑅𝐿 − 𝑐] + (1 − 𝑝)𝑓(𝐿)

which re-arranging reduces to 𝑝[(1 − 𝛼)𝑓(𝐿) + 𝑐] ≥ (1 − 𝑝)𝑅𝐿. This can be

written in the equality form as:

𝑅𝐿 = 𝑝[(1−𝛼)𝑓(𝐿)+𝑐]

(1− 𝑝) (3.1)

Equation (3.1) is the reduced form version of the borrower’s incentive

constraint precluding default, and states that the total obligation of the borrower must not exceed a multiple of the expected costs from default.32

Loan Contracts – Bank and Borrowers

The bank’s profit after tax earned from financial intermediation activities is

(𝑅𝐿 − 𝑟_𝑑𝐷 + 𝑟_𝑓𝑆)(1 − 𝜏) − ℎ(𝑝), where 𝜏is the tax rate, 𝑟_𝑑 is the rate paid on

deposits, and 𝐷 is the amount of deposits. The cost of monitoring, ℎ(𝑝), is an

increasing and convex function of the bank’s monitoring effort with ℎ/(𝑝) > 0 and

ℎ//(𝑝) > 0. An example of such a monitoring cost function is: ℎ(𝑝) = 𝑎𝑝 +1

2𝑏𝑝 2_,

where𝑎 > 0 and 𝑏 > 0 are constant, and where the cost of monitoring tends to

increase rapidly with the effort devoted to monitoring. The bank holds a safe asset,

𝑆 > 0 and earns a risk-free return, 𝑟𝑓.

32 _{In Equation (3.1) the present value of the equilibrium loan can be written as: 𝐿 =} 𝑝[(1−𝛼)𝑓(𝐿)+𝑐]

61

The bank’s balance sheet comprising the sources of funds, 𝐷, equals the

total uses of the fund, which are: the sum of loan disbursements, 𝐿; reserve

requirements, 𝑋; and the safe asset, 𝑆. This can be expressed as:

𝐷 = 𝐿 + 𝑋 + 𝑆 (3.2)

Since the reserve requirement is mandatory and a constant fraction of the total deposits, X = β𝐷, 0 < 𝛽 < 1. Incorporating X into (3.2) gives:

𝐷(1 − 𝛽) = 𝐿 + 𝑆 (3.3)

Assuming that the bank earns a return of 𝑟0 = 0 on reserves, the profit (after using the identity of balance sheet and reserve requirements as given in (3.2) and (3.3) respectively) can be expressed as 𝜋𝑏 = [𝑅𝐿 − 𝑟𝑑𝐷 + 𝑟𝑓{𝐷(1 − 𝛽) −

𝐿}](1 − 𝜏) − ℎ(𝑝), which can be rewritten as:

𝜋𝑏 = [𝑅𝐿 − {𝑟_𝑑 − 𝑟_𝑓(1 − 𝛽)}𝐷 − 𝑟_𝑓𝐿](1 − 𝜏) − ℎ(𝑝) (3.4)

This yields the bank’s objective function, where the bank maximizes profit

by choosing 𝑅, 𝐿, and 𝑝, subject to (3.1). That is, the bank offers a combination of

the loan rate 𝑅, and the loan size 𝐿, and commits to a monitoring policy 𝑝, to

maximise profit as given in (3.4). Incorporating (3.1) into (3.4), yields the objective function in reduced form:

𝜋𝑏(𝑝, 𝐿) = [𝑝[(1−𝛼)𝑓(𝐿)+𝑐]

(1− 𝑝) − {𝑟𝑑− 𝑟𝑓(1 − 𝛽)}𝐷 − 𝑟𝑓𝐿] (1 − 𝜏) − ℎ(𝑝),

where 𝜋𝑏_{(𝑝, 𝐿)is the bank’s profit function with two choice variables, 𝑝 and 𝐿. The} reduced form profit function above includes: (i) the incentive compatibility condition; (ii) the balance sheet identity; and (iii) the reserve requirement constraint.

62

The first-order conditions with respect to 𝐿 and 𝑝 for the optimum are:

𝑝(1 − 𝛼)𝑓/_{(𝐿) = 𝑟}

𝑓(1 − 𝑝), which can also be expressed as:

𝑝(1−𝛼)𝑓/(𝐿)

(1−𝑝) = 𝑟𝑓 (3.5)

and

[(1−𝛼)𝑓(𝐿)+𝑐](1−𝜏)

(1−𝑝)2 = ℎ/(𝑝) (3.6)

The incentive constraint preventing strategic default is given by:

𝑅𝐿 = 𝑝 (1−𝛼)𝑓(𝐿)+𝑐

(1− 𝑝) (3.7)

Equations (3.5) and (3.6) determine jointly the optimal loan amount (𝐿∗)

and monitoring effort (𝑝∗_{) of the bank. The optimal values in Equation (3.7) can be} substituted to solve for the optimal 𝑅∗as a function of the tax rate, technology,

costs of default, and other parameters. Equation (3.5) describes the trade-off for the optimal disbursement of the loan. The left-hand side represents the incremental productivity of the loan, while the right-hand side is the marginal cost of loan, which is the risk-free rate that the bank could have earned.

Equations (3.6) and (3.7) can be combined to derive the relationship between the loan rate (𝑅∗_{), monitoring effort (𝑝}∗_{) and the tax rate 𝜏:}

𝑅∗_𝐿∗_{(1 − 𝜏) = 𝑝ℎ}/_{(𝑝)(1 − 𝑝) (3.8)}

The left-hand side of Equation (3.8) is the bank’s marginal after-tax loan loss from a reduction in monitoring activity. The right-hand side captures the marginal savings from a reduction in monitoring activity. Equation (3.8) also captures the relationship between 𝑅∗ _{and 𝜏. We return to this relationship later} when discussing Hypothesis 4.

63

The model so far completes the borrowing side of the bank loan where the optimal borrowing rate is 𝑅∗(𝑟_𝑓, τ), the optimal loan amount issued by the bank is 𝐿∗(𝑟𝑓, τ) and the optimal probability of monitoring is 𝑝∗(𝑟𝑓, τ). Next, we discuss the deposit contracts offered by the bank under competitive market conditions.

Deposit Contracts – Bank and Depositors:

Depositors of the bank are economic agents who smooth consumption over time (as in any standard model). We assume two periods, 𝑡 = 0, 1. Depositors have

endowments of 𝑤₀ at period 0 and 𝑤₁ at period 1 with 𝑤₀ > 𝑤₁. If depositors

deposit 𝐷 with a bank and are promised a deposit rate equal to 𝑟𝑑, then the

depositors’ budget constraints are 𝑤₀ = 𝑐₀+ 𝐷 and 𝑤₁+ 𝐷𝑟_𝑑 = 𝑐₁, in each of the

two periods, 𝑡 = 0, 1, respectively, where 𝑐_𝑡 denotes the consumption of the

depositors at time t.

If the depositor’s utility function is 𝑢(𝑐₀) + 𝜃𝑢(𝑐₁), then intertemporal

maximization of utility would generate an optimal deposit function of 𝐷∗ = 𝐷∗(𝑟𝑑). For example, if the depositor has a logarithmic utility function, then the optimal deposit function is given by 𝐷∗ = 1

1+𝜃(𝑤0− 𝑤1

𝑟𝑑).

33 _{Thus for any deposit rate, 𝑟}

𝑑 offered by banks, individual depositors save 𝐷∗.

We assume that the competitive structure of the market, results in an equilibrium determination of the deposit rate where banks earn zero profit and depositors maximize utility. Proceeding with the logarithmic utility function of the

depositors, a bank’s competitive zero profit condition implies that the following condition holds for all banks:

33 _{The first order condition for a logarithmic utility function is:} 1

𝑤0−𝐷=

𝜃𝑟𝑑

𝑤1+𝑟𝑑𝐷. By rearranging, we get the

equation for 𝐷∗₌ 1 1+𝜃(𝑤0−

𝑤1

64 𝜋𝑏∗_{(𝑝, 𝐿) = [𝑝}∗ (1−𝛼)𝑓(𝐿∗)+𝑐 (1− 𝑝∗₎ − {𝑟𝑑− 𝑟𝑓(1 − 𝛽)} 1 1+𝜃(𝑤0− 𝑤1 𝑟𝑑) − 𝑟𝑓𝐿 ∗_{] (1 − 𝜏) − ℎ(𝑝}∗_{) = 0} (3.9)

where * denotes a variable set at the optimal level given by Equations (3.5) and (3.6). Equation (3.9) determines the optimal deposit rate 𝑟𝑑 = 𝑟𝑑(𝜏). Deposits are

determined by 𝐷∗ ₌ 1

1+𝜃(𝑤0− 𝑤1

𝑟𝑑(𝜏)).